A series is given with one term missing. Select the correct alternative from…

CONCEPT: When consecutive-term differences of a series do not form a simple arithmetic or geometric progression, check whether the differences themselves follow a polynomial rule in the term's position n (a common form is n3 − n2). Such a rule can be confirmed independently via finite differences: continuing to take differences of differences eventually produces a constant row — for a rule of degree k in n, that constant row appears at the k-th level of differencing the difference sequence itself.

APPLICATION:

1. Label the given terms by position: T1 = 1, T2 = 1, T3 = 5, T4 = 23, T5 = 71, T6 = 171.

2. Compute the first differences between consecutive terms: T2 − T1 = 0, T3 − T2 = 4, T4 − T3 = 18, T5 − T4 = 48, T6 − T5 = 100.

3. Check whether these differences match n3 − n2 for n = 1, 2, 3, 4, 5: 13−12=0, 23−22=4, 33−32=18, 43−42=48, 53−52=100 — all five match exactly, confirming the rule governing this series.

4. Apply the rule for the missing term, using n = 6: the required next difference is 63 − 62 = 216 − 36 = 180.

5. Add this difference to the last known term: T7 = 171 + 180 = 351.

CROSS-CHECK:

Taking differences of the given six terms directly, without assuming the n3 − n2 form: first differences 0, 4, 18, 48, 100; second differences 4, 14, 30, 52; third differences 10, 16, 22. These third differences rise by a constant 6 each step — equivalently, the fourth-level differences are 6, 6, a genuine constant row — confirming the underlying rule is degree-3 (cubic) in n, matching the concept above. Extending the pattern: the next third difference is 22 + 6 = 28, giving the next second difference 52 + 28 = 80, giving the next first difference 100 + 80 = 180, so T7 = 171 + 180 = 351 — the same result obtained above.

The missing term is 351, so the option "351" completes the series.